# Tesla Coil Spark Physics — Cheat Sheet

Everything you need to see the whole picture, in order.

---

## 1. The Spark Is a Circuit Element

A spark hanging off a topload is not magic plasma — it's a **lossy capacitive load** on a resonant circuit. It has exactly three electrical properties that matter:

- **C_mut** (mutual capacitance): coupling between topload and spark channel (~3-15 pF)
- **C_sh** (shunt capacitance): coupling between spark channel and ground (~2 pF/foot)
- **R** (resistance): the lossy part where power is dissipated (1 kohm to 100 Mohm)

These three form a bridged-T network. That's the entire circuit model of a spark.

## 2. The Phase Constraint

Because C_mut and C_sh form a capacitive divider, the impedance phase at the topload is **always more negative than -45 degrees** for typical TC geometries. You can't achieve a conjugate match. This is a topological fact, not a design flaw.

Typical impedance phase at optimum: **-55 to -75 degrees**.

## 3. The Plasma Self-Optimizes

The spark resistance R isn't fixed — it adjusts itself through heating and ionization. The key result:

**The plasma drifts toward R_opt_power = 1/(omega * C_total)** because:
- Too high R → less power → less heating → R rises further (unstable, spark dies or branches)
- Too low R → less power (past optimum) → but stronger heating prevents R from dropping much below optimum

This is the **hungry streamer principle**: the spark "eats" as much power as the circuit can deliver, automatically finding the impedance that maximizes power transfer.

## 4. Two Ways to Grow

A spark extends its length when two conditions are met:

**Condition 1 — Field threshold:** E_tip > E_propagation
The electric field at the spark tip must exceed the propagation threshold. If it doesn't, the spark stalls regardless of available power.

**Condition 2 — Energy supply:** dL/dt = P_stream / epsilon
Growth rate equals available power divided by energy cost per meter.

**Critical nuance:** E_propagation is NOT a fixed constant. In cold air, E_propagation ~ 0.5 MV/m. But at a driven leader tip, four mechanisms — UV pre-ionization, thermal pre-conditioning, residual ionization, and gas expansion — converge to dynamically reduce it. This is why QCW achieves 2+ m sparks at only 40-70 kV: the leader creates its own favorable conditions. Voltage and power are coupled limits, not independent ones (Section 4A).

The spark is always limited by whichever constraint binds first: **voltage-limited** (can't push field high enough even with dynamic threshold) or **power-limited** (can extend field but not fast enough).

## 5. Epsilon: The Central Parameter

**Epsilon (J/m)** = energy required per meter of spark growth. It varies enormously:

| Mode | Epsilon | Why |
|---|---|---|
| QCW (leader) | 5-15 J/m | Hot, efficient single channel |
| Burst (streamer) | 30-100+ J/m | Cold, branched, inefficient |

The difference is almost entirely explained by **channel type** (Section 7) and **branching** (Section 10).

## 6. The Capacitive Divider Problem

As the spark grows, C_sh increases (more conductor length to ground). This **divides down the tip voltage**:

```
V_tip = V_topload * C_mut / (C_mut + C_sh)
```

Longer spark → more C_sh → lower V_tip → weaker E_tip → harder to keep growing.

This creates **sub-linear scaling**: doubling energy does NOT double spark length. Burst mode follows L ~ sqrt(E). QCW is somewhat better (L ~ E^0.6-0.8) because leader channels have lower C_sh per unit length than branched streamers.

## 6A. The Dynamic Threshold

The capacitive divider predicts QCW sparks should stall at well under 1 m with only 40-70 kV topload. Yet 2+ m sparks are routinely achieved. The resolution: **E_propagation is not a fixed constant** — at a driven leader tip, four mechanisms converge to reduce it:

1. **UV photoionization** — corona creates seed electrons ahead of the tip
2. **Thermal pre-conditioning** — heat reduces gas density (E_breakdown proportional to N proportional to 1/T)
3. **Residual ionization** — previous streamers leave persistent electron density (~50 us decay)
4. **Gas expansion** — lower N means lower absolute field threshold

These are mutually reinforcing: more leader current drives all four harder. The result is a **coupled voltage-power limit** — power modifies the conditions that set the voltage threshold. More power → lower effective E_propagation → spark extends further at the same voltage.

But there is a floor: E_propagation can't reach zero. The capacitive divider wins eventually. The "too long" QCW regime (>25 ms) is exactly the point where even maximal pre-conditioning can't keep E_tip above the reduced threshold.

## 7. Two Kinds of Channel

This is the fork in the road that explains almost everything:

| | Streamer | Leader |
|---|---|---|
| Temperature | 300-3000 K | 5,000-20,000 K |
| Diameter | 10-100 um | 1-10 mm |
| Resistance | Very high | Low |
| Persistence | Microseconds | Seconds |
| Branching | Extensive | Minimal |
| Epsilon | High (30-100+) | Low (5-15) |
| Color | Purple/blue | White/yellow |

Streamers are cold, thin, branched, and inefficient. Leaders are hot, thick, straight, and efficient. **The entire game is getting from streamer to leader.**

## 8. The Thermal Ratchet

The transition from streamer to leader requires heating the channel past ~5000 K (through intermediate thresholds at 2000 K and 4000 K). But thin channels cool fast:

```
tau_thermal = d^2 / (4 * alpha)      alpha ~ 2e-5 m^2/s for air
```

A 100 um streamer cools in ~125 us. You have to heat it faster than it cools.

The **conductance relaxation** is asymmetric:
- Heating: tau_g = 40 us (fast — ionization responds quickly to current)
- Cooling: tau_g = 200 us (slow — recombination and thermal diffusion take longer)

This 5:1 asymmetry creates a **one-way thermal ratchet**: each RF cycle heats a little more than the previous one cooled. Over many cycles, temperature accumulates monotonically upward through the critical zone.

## 9. Frequency Matters

The ratchet only works if the RF period is much shorter than tau_thermal:

- At **400 kHz** (T_half = 1.25 us): streamer experiences ~100 RF cycles per tau_thermal. Heating is effectively continuous. Ratchet works. → **Swords.**
- At **100 kHz** (T_half = 5 us): thin streamers cool significantly between cycles. Ratchet is intermittent. → **Branchy, noisy sparks.**

The community-observed threshold: **300-600 kHz** for sword sparks. This is not about breakdown physics — it's about whether the thermal ratchet can outrun cooling.

## 10. Branching Is a Competition

Discharges branch because of **Laplacian instability** at the propagating tip (same physics as viscous fingering). Streamers branch every ~10-20 diameters.

But branches **compete** for current. The channel resistance follows a nonlinear power law:

```
R = A / I^b      b = 1.84 for TC currents (1-10 A)
```

Because b > 1, the V-I curve has **negative slope**. A branch that gets slightly more current heats up, becomes more conductive, steals more current from its neighbors. This is positive feedback — **one branch wins, the rest die.**

Competition timescale: ~120-200 us (a few tau_g).

- **Burst mode** (70-150 us pulses): too short for competition to resolve → many branches survive → bushy
- **QCW mode** (10-20 ms ramp): competition resolves in <1 ms → single dominant channel → sword
- **Pulse-skip**: intermediate — competition operates but with jitter → "sword-like but still branches"

## 11. QCW vs Burst: The Complete Picture

| | QCW | Burst |
|---|---|---|
| Voltage | 40-70 kV (!!) | 200-600 kV |
| Duration | 10-20 ms | 70-150 us |
| Frequency | 300-600 kHz | 50-200 kHz |
| Channel type | Leader | Streamer |
| Branching | Suppressed by competition | Extensive |
| Epsilon | 5-15 J/m | 30-100+ J/m |
| Spark:secondary ratio | 7-16x | 2.5-3.6x |
| Morphology | Straight sword | Bushy tree |
| Mechanism | Thermal ratchet over many ms | Brute-force high voltage |

The 15:1 voltage ratio (measured by davekni) is the single most striking number. QCW achieves leader formation at 40-70 kV because it has **time** — the ratchet accumulates thermal energy over 10-20 ms. Burst needs 200-600 kV because it must reach leader temperature in a single ~100 us pulse.

## 12. The Three Ramp Regimes

QCW ramp duration selects three distinct outcomes:

- **Too short (<5 ms):** Insufficient time for streamer-to-leader transition. Segmented, gnarly sparks.
- **Optimal (10-20 ms):** Leader forms within 1-2 ms, grows as single channel for remainder. Straight swords.
- **Too long (>25 ms):** Leader reaches voltage-limited max length (capacitive divider). Excess energy drives lateral breakouts. "Hot, fat, bushy."

## 13. Putting It All Together

The complete causal chain:

```
RF drive at frequency f
    │
    ├─→ Resonant voltage gain → V_topload
    │
    ├─→ E_tip = kappa * V_tip / L  → inception when E_tip > E_inception
    │
    ├─→ Streamer channels form (cold, branched, high R)
    │
    ├─→ Hungry streamer: R drifts toward R_opt_power
    │       │
    │       ├─→ Power delivered: P = f(V_th, Z_th, R)
    │       │
    │       └─→ Growth: dL/dt = P / epsilon
    │
    ├─→ Thermal evolution (depends on mode):
    │       │
    │       ├─→ QCW: sustained ramp → thermal ratchet → leader formation
    │       │     → branch competition selects single channel
    │       │     → low epsilon → efficient growth → sword
    │       │
    │       └─→ Burst: short pulse → no time for leader transition
    │             → branches coexist → high epsilon → bushy
    │
    ├─→ Dynamic threshold (QCW only):
    │       Leader current → UV + heat + residual ionization + expansion
    │       → E_propagation_effective drops well below cold-air value
    │       → spark extends further at lower voltage
    │       → coupled V-P limit, not independent constraints
    │
    ├─→ Capacitive divider: C_sh grows with L
    │       → V_tip decreases → E_tip drops
    │       → eventually E_tip < E_propagation_effective → stalls
    │       → sub-linear scaling: L ~ sqrt(E) for burst
    │
    └─→ Final length set by:
            min(dynamic voltage limit, energy limit, ramp duration)
```

## 14. The Numbers That Matter

| Quantity | Value | Why it matters |
|---|---|---|
| C_sh per foot | ~2 pF | Sets voltage division rate |
| R_opt_power | 10-100 kohm | Where plasma naturally sits |
| E_propagation (cold air) | 0.4-1.0 MV/m | Field floor for cold streamer growth |
| E_propagation (leader tip) | Much lower (T3) | Dynamically reduced by UV/heat/ionization |
| tau_thermal (100 um) | ~125 us | Streamer cooling timescale |
| tau_g (heating) | 40 us | Conductance response speed |
| tau_g (cooling) | 200 us | 5:1 asymmetry drives ratchet |
| Competition time | ~120-200 us | Branch winner decided |
| Burst ceiling | ~80 us | ON time saturation (Steve Ward) |
| QCW optimal ramp | 10-20 ms | Sweet spot for leader growth |
| Frequency threshold | 300-600 kHz | Below this, no swords |
| QCW voltage | 40-70 kV | 15:1 less than burst |
| da Silva b exponent | 1.84 | b > 1 → current hogging |
| Fractal dimension | ~2.2 | Streamer tree space-filling |

## 15. What We Don't Know

1. **Exact branching power division** — no validated current-sharing rule
2. **Epsilon from first principles** — still requires calibration
3. **Time-resolved impedance during QCW ramp** — never measured
4. **Spectroscopic temperature of QCW sparks** — 5000 K inferred, not measured
5. **Arc current in any QCW spark** — secondary current unmeasured
6. **How C_sh scales with branching** — qualitative only
7. **Branching fraction of epsilon** — how much energy goes to side branches vs other overhead
8. **Dynamic threshold magnitude** — how much is E_propagation reduced at a QCW leader tip?
9. **Gas temperature ahead of leader tip** — spectroscopic measurement needed